Show that four points whose position vectors are 6 i -7 j ,16 i -… - Vidyadip

Question

Show that four points whose position vectors are
$6\hat{\text{i}}-7\hat{\text{j}},16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}},3\hat{\text{i}}-6\hat{\text{k}},2\hat{\text{i}}-5\hat{\text{j}}+10\hat{\text{k}}$ are coplanar.

✓

Answer

DISCLAIMER: Given points are not coplaner.
Let A, B, C, D be the given points. The given points will be coplanar iff any one of the follewing triads of vectors are coplanar:
$\vec{\text{AB}},\vec{\text{AC}},\vec{\text{AD}};\vec{\text{AB}},\vec{\text{BC}},\vec{\text{CD}};\vec{\text{BC}},\vec{\text{BA}},\vec{\text{BD}}$ etc.
In order to show that $\vec{\text{AB}},\vec{\text{AC}},\vec{\text{AD}}$ are coplanar, we will have to show that their scaler triple
product i.e. $\Big[\vec{\text{AB}}\vec{\text{ AC }}\vec{\text{AD}}\Big]=0$
Using, $\vec{\text{PQ}}$ = Position vector of Q - position vector of P, we obtain
Now,
$\vec{\text{AB}}=(16\hat{\text{i}}-19\hat{\text{j}}-4\hat{\text{k}})-(6\hat{\text{i}}-7\hat{\text{j}})\\=10\hat{\text{i}}-12\hat{\text{j}}-4\hat{\text{k}}$
$\vec{\text{AC}}=(3\hat{\text{i}}-6\hat{\text{k}})-(6\hat{\text{i}}-7\hat{\text{j}})\\=-3\hat{\text{i}}+7\hat{\text{j}}-6\hat{\text{k}}$
and, $\vec{\text{AD}}=(2\hat{\text{i}}-5\hat{\text{j}}+10\vec{\text{k}})-(6\hat{\text{i}}-7\hat{\text{j}})\\=-4\hat{\text{i}}+2\hat{\text{j}}+10\hat{\text{k}}$
$\therefore\Big[\vec{\text{AC}}\vec{\text{ AC }}\vec{\text{AD}}\Big]=\begin{vmatrix}10&-12&-4\\-3&7&-6\\-4&2&10 \end{vmatrix}$
$=10(70+12)+12(-30-24)-4(-6+28)=84$
Thus, the given points are not coplanar.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Scalar Triple Product→Scalar Triple Product — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following functions with respect to x:
$\text{e}^{3\text{x}}\cos2\text{x}$

Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.

A trust invested some money in two type of bonds. The first bond pays 10% interest and bond pays 12% interest. The trust received 2,800 as interest. However, if trust had interchanged money in bonds, they would have got 100 less as interest. Using matrix method, find the amount invested by the trust. Which value is reflected in this question?

Find the area of the region bounded by the parabola $y = x^2$ and $y = |x|.$

Let $A = \{1, 2, 3\}$. Then, the number of equivalence relations containing $(1, 2)$ is:

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Evaluate the following intergrals:
$\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}$

Two schools $P$ and $Q$ want to award their selected students on the values of Discipline, Politeness and Punctuality. The school $P$ wants to award ₹ $x$ each, ₹ $y$ each and ₹ $z$ each for the three respective values to its $3, 2$ and $1$ students with a total award money of ₹ $1,000.$ School $Q$ wants to spend ₹ $1,500$ to award its $4, 1$ and $3$ students on the respective values $($by giving the same award money for the three values as before$).$ If the total amount of awards for one prize one each value is ₹ $600,$ using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{3}}\frac{\cos\text{x}}{3+4\sin\text{x}}\text{ dx}$

Solve the following differential equation:
$(\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=2\sin^2\text{x}\cos\text{x}$